New results on affine invariant points (Q2627986)

An \textit{affine invariant point} is a continuous map \(p:\mathcal{K}_d\longrightarrow{\mathbb R}^d\) from the family of \(d\)-dimensional convex bodies (endowed with the Hausdorff metric) to \({\mathbb R}^d\) (with the Euclidean norm) such that, for every \(C\in\mathcal{K}_d\) and every invertible affine map \(T\), we have \(p(T(C))=T(p(C))\). For instance, the center of gravity or the Santaló point are examples of affine invariant points. In 1963, B. Grünbaum posed the question whether the symmetry structure of a convex body can be related to the set of affine invariant points. In the paper under review the author answers in the affirmative the above question, and shows that given a convex body \(C\), the set of affine invariant points of \(C\) is \[ \{x\in{\mathbb R}^d:Tx=x\text{ for every \(T\) affine linear s.t. }T(C)=C\}. \] As a consequence of this characterization result, the author provides a new short proof of the known fact that the set of affine invariant points of a convex body has infinite dimension. Moreover, he proves that there exists an open and dense subset of the set of affine invariant points whose elements have no dual; here it is said that an affine invariant point \(q\) is \textit{dual} to an affine invariant point \(p\) if and only if \(q((C-p(C))^{\circ})=0\).